#### Vol. 16, No. 1, 2022

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Gromov–Witten theory of $[\mathbb{C}^2 / \mathbb{Z}_{n+1}] \times \mathbb{P}^1$

### Zijun Zhou and Zhengyu Zong

Vol. 16 (2022), No. 1, 1–58
##### Abstract

We compute the relative orbifold Gromov–Witten invariants of $\left[{ℂ}^{2}∕{ℤ}_{n+1}\right]×{ℙ}^{1}$ with respect to vertical fibers. Via a vanishing property of the Hurwitz–Hodge bundle, 2-point rubber invariants are calculated explicitly using Pixton’s formula for the double ramification cycle, and the orbifold quantum Riemann–Roch. As a result parallel to its crepant resolution counterpart for ${\mathsc{𝒜}}_{n}$, the GW/DT/Hilb/Sym correspondence is established for $\left[{ℂ}^{2}∕{ℤ}_{n+1}\right]$. The computation also implies the crepant resolution conjecture for the relative orbifold Gromov–Witten theory of $\left[{ℂ}^{2}∕{ℤ}_{n+1}\right]×{ℙ}^{1}$.

##### Keywords
relative orbifold GW theory, GW/DT correspondence, crepant resolution conjecture
Primary: 14N35